Recursive and trellis-based feedback reduction for MIMO-OFDM with rate-limited feedback

ABSTRACT

Techniques are provided for reducing feedback while maintaining performance in a MIMO-OFDM system. The disclosed techniques employ finite-rate feedback methods that uses vector quantization compression. The disclosed methods/techniques generally involve: receiving a plurality of symbols from a plurality of sub-carriers at a receiver; selecting a plurality of indices of codewords corresponding to a codebook of pre-coding weighting matrices for the sub-carriers based on vector quantization compression of the codewords; and transmitting the selected indices over a wireless channel to a transmitter. Finite state vector quantization feedback makes use of a finite state vector quantizer (FSVQ), which is a recursive vector quantizer (VQ) with a finite number of states. In finite state vector quantization feedback, optimal precoding matrices (beamforming vectors) are selected sequentially across subcarriers. In a trellis-based feedback method, the optimal precoding matrices are selected at the same time for all subcarriers by searching for the optimum choice of matrices along a trellis using the Viterbi algorithm (dynamic programming).

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. provisional patentapplication No. 60/737,815, filed Nov. 17, 2005, the disclosure of whichis incorporated herein by reference in its entirety.

STATEMENT OF GOVERNMENT INTERESTS

The research giving rise to the present invention was sponsored by theOffice of Naval Research under Contract Number N00014-04-1-013.Accordingly, the government has certain interests in the presentinvention.

TECHNICAL FIELD

The present invention relates to digital wireless communication and,more particularly, to a method for reducing feedback while improvingcommunications performance in an adaptive MIMO-OFDM system.

BACKGROUND

The wireless channel in a wireless communication system constitutes ahostile propagation medium. This medium suffers from fading, which iscaused by the destructive addition of multiple replicas of a transmittedsignal which are detected by a receiver from multiple paths. Anotherproblem with wireless data transmission is interference from otherusers. One approach to combating fading and interference is to provide areceiver with several identical replicas of a transmitted signal. Thisis accomplished by diversity techniques. One such technique is antennadiversity, in which an array of antennas is deployed at the transmitside and/or the receive side of a wireless link. Another name for asystem which employs antenna diversity at both the transmit side andreceive side of a wireless link is a multiple-input multiple-output(MIMO) system.

FIG. 1 depicts a typical MIMO system, generally indicated at 10, whichis known in the prior art. Digital data 12 is encoded by an encoder 14using one of several encoding techniques, such as quadrature amplitudemodulation (QAM). The encoded data is demultiplexed into several datastreams by a demultiplexer 16. The demultiplexer 16 feeds the multipledata streams to a plurality of modulators 18 _(a)-18 _(Nt) usingorthogonal frequency division multiplexing (OFDM) to be discussedhereinbelow. The modulated signals are then transmitted through thewireless medium simultaneously via N_(t) antennas 20 a-20 _(Nt). Whiletraveling through the wireless medium, some of the signals transmittedfrom the antennas 20 a-20 _(Nt) may reflect from obstructions 22, 24,such as buildings, cars, trees, and the like. Signals transmitteddirectly from all of the sending antennas 20 a-20 _(Nt) and indirectlyfrom the obstructions 22, 24 are received at N_(r) receive antennas 26a-26 _(Nr). A MIMO receiver 28 demodulates, multiplexes, and decodes theseveral received data streams into a single received data stream.Designating the signal received at each of the receive antennas 26 a-26_(Nr) as y_(j), and the signals transmitted at the sending antennas 20a-20 _(Nt) as x_(i), the signals received at each of the receiveantennas 26 a-26 _(Nr) can be represented as a set of linear equationswherein “h” is the signal weight, as follows:y ₁ =h ₁₁ x ₁ +h ₁₂ x ₂ + . . . +h _(1N) _(t) x _(N) _(t)y ₂ =h ₂₁ x ₁ +h ₂₁ x ₂ + . . . +h _(2N) _(t) x _(N) _(t)...y _(Nr) =h _(N) _(r) ₁ x ₁ +h _(N) _(r) ₂ x ₂ + . . . +h _(N) _(t) _(N)_(r) x _(Nt)

As can be seen from the above equations, in making their way from thesending antennas 20 a-20 _(Nt) to the receive antennas 26 a-26 _(Nr),the independent signals, x₁ through x_(Nt), are all combined.Traditionally, this “combination” has been treated as interference. Bytreating the channel as a matrix, however, the independent transmittedstreams, x_(i), can be recovered by estimating the individual channelweights h_(ij). The transmitted signals x_(i) and the received signalsy_(j) can be collected into vectors, x and y of dimensions N_(t)×1 andN_(r)×1, respectively, and the channel weights h_(ij) can be collectedinto a channel matrix H of dimensions N_(r)×N_(t). The channelinput-output relationship in matrix-vector form can be expressed as:y=Hx+vwhere v is a vector of channel noise to be discussed hereinbelow. Havingestimated H, one can solve for the values of the transmit vector x bymultiplying the receive vector y by the inverse of H and subtracting vtherefrom.

Because multiple data streams are transmitted in parallel from differentantennas, there is a linear increase in throughput with every pair ofantennas added to the system. In current wireless communication schemes,there is not only a need to increase throughput, but also a need toimprove the quality of the received signals. Because of reflections fromdifferent obstructions 22, 24, sometimes the reflected signals add up inphase and sometimes they add up out of phase causing a “fade”. A fadecauses the received signal strength to fluctuate constantly. Differentsub-channels (the transmitted signals) are distorted differently, whichleads to a sub-channel becoming frequency selective. As throughput (datarate) increases, frequency selectivity also increases. Systems employingorthogonal frequency division multiplex (OFDM) modulation convertfrequency selective channels into a set of parallel flat-fadingsub-channels, thus enabling low complexity equalization. The“orthogonal” part of the name refers to a property of sub-channels inwhich the frequencies in each sub-band are integer multiples of afundamental frequency. This ensures that even though the sub-channelsoverlap, they do not interfere with each other, thereby removing thefrequency selectivity and thus increasing spectral efficiency. Thewedding of MIMO and OFDM for high speed applications combines highthroughput with high spectral efficiency.

The operations performed on a transmit side of a datastream of a blockof symbols of length N as it passes through an OFDM modulator includethe steps of: encoding the block of symbols using an encoding schemesuch as quadrature amplitude modulation (QAM); passing the encodedsymbols through a serial-to-parallel converter; performing an InverseFast Fourier Transformation (IFFT) on the parallel data; prepending tothe parallel data a cyclic prefix (CP) of length L_(CP)≧L containing acopy of the last L_(CP) samples of the parallel-to-serial convertedoutput of the N-point IFFT; and passing the data through theparallel-to-serial converter. The length of the cyclic prefix (CP) beinggreater than or equal to the length of the discrete-time basebandchannel impulse response (i.e., L_(CP)>L) guarantees that thefrequency-selective MIMO fading channel decouples into a set of parallelfrequency-flat MIMO fading channels. The symbols are converted back toanalog form and transmitted from transmit antennas into the wirelessmedium.

On the receive side of the wireless medium, an antenna receives anOFDM-modulated signal and passes the signal through an OFDMde-modulator. The OFDM de-modulator performs the following operations:stripping the cyclic prefix; converting the serial data to parallelform; performing an N-point Fast Fourier Transformation (FFT) on thedata, converting the parallel data back to serial form, and decoding thedata (e.g., via a QAM-decoder).

To improve data transmission and error performance still further,finite-rate transmit beamforming is applied to multiple data symbols xto be transmitted for each of p sub-carriers. In finite-rate transmitbeamforming, some of the data bits received at the receiver 28 are fedback to the transmitter/modulators 18 a-18 _(Nt) so that the transmittercan adapt to changing channel conditions via beamforming weights appliedto the signal to be transmitted. In a multi-antenna wirelesscommunication system having N_(t) transmit-antennas and N_(r)receive-antennas, each transmit antenna employing OFDM using N_(c)subcarriers, the fading channel between the μ-th transmit-antenna andthe υ-th receive-antenna is assumed to be frequency-selective buttime-flat, and is described by a linear filter with LL+1 taps, asfollows:h _(υμ)(n):={h _(υμ)(n;0), . . . , h _(υμ)(n;LL)},where n is the OFDM symbol index and LL is the channel order. Thechannel impulse response includes the effects of transmit-receivefilters, physical multipath, and relative delays among antennas. With pdenoting the OFDM subcarrier index, the frequency response between theμ-th transmit-antenna and the υ-th receive-antenna on the p-thsubcarrier is:${{H_{\upsilon\mu}\left\lbrack {n;p} \right\rbrack} = {\sum\limits_{l = 0}^{LL}\quad{{h_{\upsilon\quad\mu}\left( {n;l} \right)}{\mathbb{e}}^{- j^{2\pi\quad{{pl}/N_{c,}}}}}}},{p = 0},\ldots\quad,{N_{c} - 1.}$At the p-th subcarrier of the n-th OFDM symbol, by collecting thetransmitted symbols across N_(t) transmit antennas in an N_(t)×1 vectorx[n;p], and the received symbols across N_(r) receive-antennas in anN_(r)×1 vector y[n;p], the channel input-output relationship on the p-thsubcarrier is:Y[n;p]=H[n;p]x[n;p]+v[n;p],where v[n;p] is additive white Gaussian noise (AWGN) with each entryhaving a variance with each entry having variance N₀ and H[n;p] is theN_(r)×N_(p) channel matrix with the (υ,μ)-th entry being H_(υμ)(n;p).

With finite rate transmit beamforming, an information symbol s[n;p] ismultiplied by an N_(t)×1 beamforming vector w[p] to formx[n;p]=w[p]s[n;p], which is then transmitted through the p-thsub-carrier of the OFDM system. The input-output relationship on thep-th sub-carrier can be expressed as:y[n;p]=H[p]w[p]s[n;p]+v[n;p]Based on feedback, the transmitter seeks to match the beamforming vectorw[p] to the channel H[p] to improve system performance.

If the transmitter has perfect knowledge of H[p], the optimalbeamforming vector will be the eigenvector of H^(H)[p] H[p], whereH^(H)[p] is the Hermitian transpose of H[p], corresponding to thelargest eigenvalue to maximize the signal to noise ratio (SNR) on eachsub-channel, where the SNR is designated as γ[p], and:${\gamma\lbrack p\rbrack} = {\frac{E_{S}}{N_{0}}{{{H\lbrack p\rbrack}{w\lbrack p\rbrack}}}^{2}}$where E_(s) is the average energy per symbol s[n;p] and ∥*∥ denotes atwo-norm of a vector or a matrix. Assuming a maximum ratio combining(MRC) receiver, the received symbols ŝ[n;p] are:ŝ[n;p]=w^(H)[p]H^(H)[p]y[n;p]where w^(H)[p] is the Hermitian transpose of w[p] and H^(H)[p] is theHermitian transpose of H[p].

Finite rate transmit beamforming satisfies the condition that channelbehavior is known to both the receiver and transmitter. This behavior isrepresented by the matrix H[p], which can be estimated by the receiver,which has knowledge of the effects of the wireless channel. This meansthat the receiver would have to estimate H[p] for each OFDM frequencychannel, and send all of this information back to the transmitter. Thisinformation is ancillary data that is not part of the informationtransmitted. Thus, it is desirable that the amount of bandwidthdedicated to feedback information be kept to a minimum. In fact, thetransmission of a matrix is an expensive operation, since a matrix hasmany elements (the square of the dimension).

A technique known as finite rate feedback can be employed to minimizethe data to be transmitted from the receiver to the transmitter, yetmaximize the knowledge gained by the transmitter about the channel toimprove beamforming. One version of finite rate feedback is “persubcarrier feedback.” In per subcarrier feedback, feedback is doneseparately on all subchannels. Assuming that B₁ feedback bits areavailable per subcarrier, the transceiver will need a codebook CB ofsize 2^(B) ¹ , which is a collection of beamforming vectors {w₁, . . . ,w₂ _(B) ₁}. It is assumed that the codebook CB is the same acrosssubcarriers. The beamforming vector is chosen at the receiver tomaximize γ[p] at the p-th subcarrier to be:${w^{opt}\lbrack p\rbrack} = {\arg\quad{\max\limits_{w = W}{{{{H\lbrack p\rbrack}w}}^{2}.}}}$The index of w^(opt)[p] will be fed to the transmitter B₁ feedback bits.The transmitter then switches to w^(opt)[p] after finding w^(opt)[p] viathe index in its own codebook. Unfortunately, with finite rate feedback,N_(c)B₁ bits need to be fed back to the transmitter, which is a largenumber of bits considering that N_(c) is usually large.

Another technique known in the art is the one employed by Choi and Heathknown as interpolation, which is described in the “Interpolation BasedTransmit Beamforming for MIMO-OFDM with Limited Feedback,” in Proc. OfInt. Conf. on Communications, Paris, France, June 2004, vol. 1, pp.249-253, which is incorporated herein by reference. In the interpolationtechnique, N_(c) subcarriers are split into N_(g) groups withN_(c)/N_(g) consecutive subcarriers per group. Only {w_(opt)[lN_(g)]}{w_(opt)[lN_(g)]}_(l = 0)^(N_(e)/N_(g)⁻¹)bits will be fed back to the transmitter, and the rest of thesubcarriers rely on the following interpolation:${w\left\lbrack {{lN}_{g} + k} \right\rbrack} = \frac{\begin{matrix}{{\left( {1 - {k/N_{g}}} \right){w^{opt}\left\lbrack {lN}_{g} \right\rbrack}} +} \\{{{\mathbb{e}}^{{j\theta}_{l}}\left( {k/N_{g}} \right)}{w^{opt}\left\lbrack {\left( {l + 1} \right)N_{g}} \right\rbrack}}\end{matrix}}{\begin{matrix}{{{\left( {1 - {k/N_{g}}} \right){w^{opt}\left\lbrack {lN}_{g} \right\rbrack}} +}} \\{{{{\mathbb{e}}^{{j\theta}_{l}}\left( {k/N_{g}} \right)}{w^{opt}\left\lbrack {\left( {l + 1} \right)N_{g}} \right\rbrack}}}\end{matrix}}$where θ₁ is chosen from a finite set {e^(jn2π/P)}_(n=0) ^(P-1). Thefeedback required for the interpolation method is(N_(c)/N_(g))(B₁+log₂P) bits, which is a significant improvement overthe finite rate feedback technique. Unfortunately, the interpolationtechnique suffers from “diversity loss,” in which bit error rate (BER)levels off as the signal-to-noise ratio (SNR) increases.

Thus, despite efforts to date, a need remains for methods that areeffective in reducing feedback in an adaptive MIMO-OFDM system whilemaintaining performance. These and other needs are satisfied by themethods/techniques described herein.

SUMMARY OF THE DISCLOSURE

The present invention overcomes the disadvantages and shortcomings ofthe prior art discussed above by providing a method for reducingfeedback in a MIMO-OFDM system. The present invention employs a feedbackmethod that uses vector quantization compression to limit the size offeedback data required to be sent to a transmitter for optimizingcommunications between MIMO-OFDM devices. In exemplary embodiments ofthe present invention, the following steps are employed: (i) receiving aplurality of symbols from a plurality of sub-carriers at a receiver;(ii) selecting a plurality of indices of codewords corresponding to acodebook of pre-coding weighting matrices for the sub-carriers based onvector quantization compression of the codewords; and (iii) transmittingthe selected indices over a wireless channel to a transmitter.

Vector quantization compression techniques for reducing feedback whilemaintaining performance may be advantageously employed according to thepresent disclosure because subchannel responses across OFDM subcarriersare correlated. Vector quantization compression generally takes twoforms: (i) finite state vector quantization feedback and (ii)trellis-based feedback. As demonstrated herein, either of the foregoingvector quantization compression approaches may be employed to advantageaccording to the present disclosure.

Finite state vector quantization feedback makes use of a finite statevector quantizer (FSVQ), which is a recursive vector quantizer (VQ) witha finite number of states. In finite state vector quantization feedback,optimal precoding matrices (beamforming vectors) are selectedsequentially across subcarriers. After selecting the first precodingmatrix from a codebook of a certain size, subsequent precoding matricesare selected from a smaller time-varying codebook per subcarrierdepending on prior decisions. In a trellis-based feedback method, theoptimal precoding matrices are selected at the same time for allsubcarriers by searching for the optimum choice of matrices along atrellis using the Viterbi algorithm (dynamic programming).

The finite state vector quantization feedback and trellis-based feedbacktechniques for reducing feedback while maintaining performance asdescribed herein are transmission schemes which are applicable totransceivers of different design, including transmitters which useadaptive (finite rate per subcarrier) beamforming, precoded spatialmultiplexing, or precoded orthogonal space-time block codes (STBC).Simulation results demonstrate that the trellis-based method outperformsan interpolation method which incurs diversity loss at highsignal-to-noise ratio.

Further features and advantages of the present invention will appearmore clearly on a reading of the following detailed description of thepresent invention, with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention, reference ismade to the following detailed description of the invention consideredin conjunction with the accompanying drawings, in which:

FIG. 1 is a block diagram of an MIMO-OFDM system as is known in theprior art;

FIG. 2 is a block diagram of a MIMO-OFDM system using vectorquantization feedback, constructed in accordance with an exemplaryembodiment of the present invention;

FIG. 3 is a graphical illustration of simple vector quantization for atwo-dimensional space;

FIG. 4 is a graphical illustration of a finite state vector quantization(FSVQ) method of feedback reduction used in accordance with an exemplaryembodiment of the present invention;

FIG. 5 is a flow chart showing processing steps of the method ofillustrated in FIG. 4;

FIG. 6 is a graphical illustration of a trellis-based method of feedbackreduction used in accordance with another exemplary embodiment of thepresent invention;

FIG. 7 is a flow chart showing processing steps of the methodillustrated in FIG. 6;

FIG. 8 is graph of bit error rates (BER) versus average signal-to-noiseratio (SNR), comparing the performance of recursive feedback reductionusing finite state vector quantization (FSVQ) for different numbers offeedback bits per codeword;

FIG. 9 is graph of bit error rates (BER) versus average signal-to-noiseratio (SNR), comparing the performance of trellis-based feedbackreduction for different numbers of feedback bits per codeword;

FIG. 10 is graph of bit error rates (BER) versus average signal-to-noiseratio (SNR), comparing the performance of transmit beamforming feedbackfor the per subcarrier method, the trellis method, and interpolationmethod;

FIG. 11 is graph of bit error rates (BER) versus average signal-to-noiseratio (SNR), comparing the performance of precoded spatial multiplexingfeedback for the per subcarrier method, the trellis method, andinterpolation method; and

FIG. 12 is graph of bit error rates (BER) versus average signal-to-noiseratio (SNR), comparing the performance of precoded orthogonal space-timeblock codes (OSTBC) feedback for the per subcarrier method, the trellismethod, and interpolation method.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

The present invention may be used in conjunction with a MIMO-OFDM systemto reduce feedback while enhancing communications. The disclosed methodsand techniques of the present invention employ a feedback method thatuses vector quantization compression to minimize the amount of feedbackdata sent to the transmitter for improving communications betweenMIMO-OFDM systems. In general, the present invention involves the stepsof: receiving a plurality of symbols from a plurality of sub-carriers ata receiver; selecting a plurality of indices of codewords correspondingto a codebook of pre-coding weighting matrices for the sub-carriersbased on vector quantization compression of the codewords; andtransmitting the selected indices over a wireless channel to thetransmitter.

With initial reference to FIG. 2, a block diagram of a MIMO-OFDM systemusing a vector quantization feedback arrangement is provided, generallyindicated at 30, and constructed in accordance with an exemplaryembodiment of the present invention. Symbols “s” are adjusted by a spacetime encoder 32, and represent units of data to be transmitted over aMIMO-OFDM communications system. The space time encoder 32 can apply oneof a plurality of weighting techniques, including adaptive (finite rateper subcarrier) beamforming, spatial multiplexing, or precodedorthogonal space-time block codes (STBC). The weighted symbols arefurther processed by a MIMO-OFDM modulator 34, transmitted through awireless medium via transmit antennas 36 (i.e., antennas 1 throughN_(t)), received by receive antennas 38 (i.e., antennas 1 throughN_(r)), demodulated by a MIMO-OFDM demodulator 40, and detected assymbols “ŝ” by symbol detectors 42 a-42 n.

The MIMO-OFDM modulator 34, transmit antennas 36, receive antennas 38,and MIMO-OFDM demodulator 40 have a construction similar to the onepreviously discussed in connection with the prior art of FIG. 1. Due tolimited feedback, the space time encoder 32 cannot take on arbitraryvalues. In order to generate weighted symbols at the space time encoder32 with sufficient performance, a finite number of bits representing theindex of a codeword corresponding to an optimal weighting vector are fedback to the space time encoder 32 via a feedback generator 44. Thefeedback generator 44 could implement finite-state or trellis-basedvector quantization techniques. At the space time encoder 32, the indexrepresented by the codeword selects the best weighting vector from thecodebook.

Finite rate beamforming, already discussed with reference to FIG. 1, isreally a special case of a more general form of feedback, in which, toincrease the transmission rate, instead of transmitting one symbol persubcarrier, multiple (N_(s)) symbols can be transmitted in parallel, atechnique known as precoded spatial multiplexing. In precoded spatialmultiplexing, at the p-th subcarrier, N_(s) information symbols areformed into an N_(s)×1 vector s[n;p]. The symbol vector s[n;p] can beprecoded by a matrix T[p] of size N_(t)×N_(s) to form a transmittedblock x[n;p]=T[p]s[n;p]. The channel input-output relationship becomes:y[n;p]=H[p]T[p]s[n;p]+v[n;p]where s[n;p] reduces to s[n;p], or one symbol per subcarrier, whenN_(S)=1. T[p] can be drawn from a finite-size codebook

:={T₁, . . . , T_(N)}.

The receiver structure used for receiving the ŝ[n;p] can be a linearminimum-mean square-error (MMSE) receiver. The linear MMSE receiverapplies element-wise symbol detection in the symbol detectors 42 a-42 nsuch that a demodulated block ŝ[n; p]=G^(mmse)y[n; p], where:G ^(mmse) =[T ^(H) [p]H ^(H) [p]H[p]T[p]+(N ₀ /E _(S))I _(K)]⁻¹ T ^(H)[p]H ^(H) [p]where I_(K) is an identity matrix of size K, and [*]^(H) is a Hermitiantranspose.

A third type of transmission scheme used in conjunction with the presentinvention applies precoded orthogonal space-time block codes (STBC) oneach OFDM subcarrier. As an illustrative example of precoded orthogonalSTBC, an Altamouti code is used. On each subcarrier, a 2×2 Alamouti codematrix is constructed, which is then precoded by a N_(t)×2 matrix T[p]to obtain the transmitted blocks x[2i;p] and x[2i+1;p] for twoconsecutive OFDM symbols. Specifically, with two symbols s[2i;p] ands[2i+1;p], the transmitter constructs:$\left\lbrack {{x\left\lbrack {{2n};p} \right\rbrack},{x\left\lbrack {{{2n} + 1};p} \right\rbrack}} \right\rbrack = {{T\lbrack p\rbrack}\begin{bmatrix}{s\left\lbrack {{2n};p} \right\rbrack} & {- {s^{*}\left\lbrack {{{2n} + 1};p} \right\rbrack}} \\{s\left\lbrack {{{2n} + 1};p} \right\rbrack} & {s^{*}\left\lbrack {{2n};p} \right\rbrack}\end{bmatrix}}$Correspondingly, the received vectors in two consecutive time slots are:$\begin{matrix}{\left\lbrack {{y\left\lbrack {{2n};p} \right\rbrack},{y\left\lbrack {{{2n} + 1};p} \right\rbrack}} \right\rbrack = {{H\lbrack p\rbrack}{T\lbrack p\rbrack}}} \\{\begin{bmatrix}{s\left\lbrack {{2n};p} \right\rbrack} & {- {s^{*}\left\lbrack {{{2n} + 1};p} \right\rbrack}} \\{s\left\lbrack {{{2n} + 1};p} \right\rbrack} & {s^{*}\left\lbrack {{2n};p} \right\rbrack}\end{bmatrix} +} \\{\left\lbrack {{v\left\lbrack {{2n};p} \right\rbrack},{v\left\lbrack {{{2n} + 1};p} \right\rbrack}} \right\rbrack}\end{matrix}$Similar to precoded spatial multiplexing, T[p] can be drawn from afinite-size codebook

:={T₁, . . . , T_(N)}. The same codebook for precoded spatialmultiplexing can be applied here.

Reducing feedback while maintaining performance is made possible by theobservation that subchannel responses across OFDM subcarriers arecorrelated. If the subchannel responses are correlated, then compressiontechniques, such as vector quantization, can be employed. Forillustrative purposes, simple vector quantization for a two-dimensionalspace is depicted in FIG. 3. Codewords 48 are represented by dottedcircles enclosed within regions 50. The codewords 48 are calledcodevectors and the regions 50 are called encoding regions. The set ofall codevectors is called the codebook. The codeword 48 representing theprecoding matrix T is chosen for all possible values of T that fallwithin the region 50, which are in the neighborhood of the codeword 48.Although it is desirable to feedback the ideal precoding matrix T,designated as 52, the best possible choice within the codebook that canbe fed back to the receiver and still be within the codebook of T isaccomplished by choosing the codeword 48. If there are 2^(B) ¹ totalpossible codewords in the codebook for T, then B₁ bits can be feed backto the transmitter from the receiver.

The criterion for choosing an optimal precoding matrix T per subcarrierdepends on the specific type of vector quantization used. In oneembodiment of the present invention, a finite state vector quantizer(FSVQ) is used. An FSVQ is a recursive vector quantizer with a finitenumber of states. A recursive vector quantizer is a vector quantizerwith memory, where the vector quantizer output depends not only on thecurrent input, but also on prior inputs. Using state variable tosummarize the influence of the past on the current operation of thevector quantizer, recursive vector quantization can be effectivelydescribed by state transition and state-dependent encoding. In thepresent invention, finite state vector quantization (FSVQ) is applied tothe precoding matrices T by choosing the optimal precoding matrix from acodebook sequentially across subcarriers from p=0 to p=N_(c)−1, where pis treated as a virtual time index for transitions between states.

Referring now to FIG. 4, finite state vector quantization (FSVQ) appliedto the choice of precoding matrices T is illustrated graphically.Initially, a codebook {T_(i)} has a total of 2^(B) ¹ codewords which canbe represented by B₁ bits (region 54). The next codeword (for the nextsubchannel) is chosen from a smaller codebook of size 2^(B) ² whichcontain the nearest neighbors to the previous chosen codeword, which canbe represented by B₂ bits where B₂<B₁ (region 56). The neighbors ofT_(i) are the set of codewords which minimize the chordal distance fromT_(i), where chordal distance from T_(i) to T_(j) is defined as:${d_{c}\left( {T_{i},T_{j}} \right)} = {\frac{1}{\sqrt{2}}{{{{T_{i}T_{i}^{\mathcal{H}}} - {T_{j}T_{j}^{\mathcal{H}}}}}_{F}.}}$where ∥*∥_(F) is a Frobenius norm of a matrix. As shown in FIG. 4, theprocess progresses sequentially from the codeword 58 in region 54, tocodeword 60 in smaller region 56, to codeword 62 in region 64, tocodeword 66 in region 68, etc., until all subcarrier indices p have beentraversed. Notice that subsequent regions (e.g., region 68) also containthe previous choice of codeword (e.g. codeword 62).

Referring now to FIG. 5, a flow chart 70 of the steps used to calculatethe optimal T[p] using finite state vector quantization (FSVQ) ispresented. At step 72, a list of a total of 2^(B) ¹ states isconstructed from a predetermined codebook

represented by B₁ bits. Each state ξ_(i) is characterized by oneprecoding matrix T_(i). At step 74, the optimal codeword is determinedfrom the codebook

and the state ξ[0] is based on:${T^{opt}\lbrack p\rbrack} = {\arg\quad{\max\limits_{{\mathcal{T}{\lbrack p\rbrack}} \in \mathcal{T}}{{{BER}\lbrack p\rbrack}.}}}$where T^(opt)[p] is the precoded matrix which minimizes the bit errorrate BER[p], at the p-th sub-carrier. Then,${{BER}\lbrack p\rbrack} = {{\frac{1}{N_{g}}{\sum\limits_{k = 1}^{N_{a}}\quad{{\phi\left( {\gamma_{k}^{mmse}\lbrack p\rbrack} \right)}.{\gamma_{k}^{mmse}\lbrack p\rbrack}}}} = {\frac{E_{g}/N_{0}}{\left\lbrack {{{T^{\mathcal{H}}\lbrack p\rbrack}{H^{\mathcal{H}}\lbrack p\rbrack}{H\lbrack p\rbrack}{T\lbrack p\rbrack}} + {\left( {N_{0}/E_{g}} \right)I_{K}}} \right\rbrack_{k,k}^{- 1}} - 1.}}$${\phi(\gamma)} = {\frac{1}{\log_{2}\sqrt{M}}{\sum\limits_{k = 1}^{\log_{o}\sqrt{M}}\quad{\frac{1}{\sqrt{M}}{\sum\limits_{t = 0}^{{{({1 - 2^{- k}})}\sqrt{M}} - 1}\quad{\times \left\{ {\left( {- 1} \right)^{\lfloor\frac{t \cdot o^{k - 1}}{\sqrt{M}}\rfloor}{\left( {2^{k - 1} - \left\lfloor {\frac{i \cdot 2^{k - 1}}{\sqrt{M}} + \frac{1}{2}} \right\rfloor} \right) \cdot 2}{Q\left( {\left( {{2i} + 1} \right)\sqrt{\frac{3}{M - 1}\gamma}} \right)}} \right\}}}}}}$${Q(x)} = {\int_{x}^{\infty}{\frac{1}{\sqrt{2\pi}}{\mathbb{e}}^{{- t^{2}}/2}\quad{\mathbb{d}t}}}$where p=0 and M is the size of a square quadrature-amplitude-modulation(QAM) “constellation.” At step 76, the value of p is incremented toevaluate the next (p-th) subcarrier. At step 78, a reduced-size codebook

_(i) is constructed which is a subset of

of size 2^(B) ² codewords, B₂<B₁, based on the previous state ξ[p−1],where

[p−1] stands for the current codebook associated with the state ξ[p−1]known at subcarrier p of the previous subcarrier p−1. The codewords inthe codebook

_(i) are determined to be the collection of codewords nearestT_(i)=T^(opt)[p−1]. The set of codewords nearest T_(i) minimize thechordal distance, defined as:${d_{c}\left( {T_{i},T_{j}} \right)} = {\frac{1}{\sqrt{2}}{{{{T_{i}T_{i}^{\mathcal{H}}} - {T_{j}T_{j}^{\mathcal{H}}}}}_{F}.}}$At step 80, the optimum precoding matrix T^(opt)[p] at subcarrier p isdetermined from:${{T^{opt}\lbrack p\rbrack} = {\arg\quad{\min\limits_{{T{\lbrack p\rbrack}} \in \quad{\mathcal{T}{\lbrack{p - 1}\rbrack}}}{{BER}\lbrack p\rbrack}}}},$Specifying T^(opt)[p] requires B₂ bits, when

[p−1] is available. The next state ξ[p] on the p-th subcarrier isdetermined to beξ[p]=ξ_(j),if T^(opt)[p]=T_(j).At step 82, if the current subcarrier p is not the last subcarrier p,then steps 76, 78, and 80 are repeated. If the last subcarrier has beenreached, then at step 84, B₁+(N_(C)−1)B₂ bits are fed back to thetransmitter, and correspond to the indices of the optimum codewordsselected. Then, at step 86, the transmitter specifies the optimalcodewords for all subcarriers based on the feedback bits.

In an alternative embodiment of the present invention, the precodingmatrices are chosen based on trellis-based feedback encoding. Intrellis-based feedback encoding, the decision can be made at timep=N_(c)−1 to specify the optimal codeword indices for all subcarriers atonce, instead of sequentially as in finite state vector quantization(FSVQ). Referring now to FIG. 6, trellis-based feedback encoding isillustrated graphically. All possible transition states are plottedalong the vertical axis, while all the subcarriers are plotted along thehorizontal axis. From all of the possible paths from subcarrier 0 tosubcarrier N_(C)−1, the optimal path through the trellis, designated atreference 98, is chosen to minimize the average bit error rate. Thispath is determined by means of the Viterbi algorithm (i.e., by dynamicprogramming).

Referring now to FIG. 7, an exemplary flow chart, designated at 100, ofthe steps used to calculate the optimal T_(i) using trellis-basedfeedback encoding is presented. At step 102, a trellis is constructed byspecifying a state transition table where all possible transition statesare plotted along the vertical axis, and all the subcarriers “p” areplotted along the horizontal axis. There are total of 2^(B) ¹ statesfrom a predetermined codebook

represented by B₁ bits. Each state ξ_(i) is chararacterized by oneprecoding matrix T_(i). At step 104, for each state ξ_(i), 2^(B) ²neighbor states are defined and are denoted by the functionNeighbor(ξ_(i), j) for j=0, . . . , 2^(B) ² −1. The next state ξ[p] isdefined as neighbor(ξ[p−1],j). The neighbor states are defined byarranging the codewords T_(i), which is the collection of 2^(B) ²codewords, B₂<B₁, from

that are closest to T_(i) in descending order of the chordal distancesrelative to T_(i) where the chordal distance is defined as:${d_{c}\left( {T_{i},T_{j}} \right)} = {\frac{1}{\sqrt{2}}{{{{T_{i}T_{i}^{\mathcal{H}}} - {T_{j}T_{j}^{\mathcal{H}}}}}_{F}.}}$Each state ξ[p−1] at the (p−1)-th subcarrier can only transit to one ofthe 2^(B) ² neighbor states on the p-th subcarrier. At step 106, abranch metric is determined from state ξ|p−1∥toξ[p] for all subcarriers“p” as:${{Metric}\left( {{\xi\left\lbrack {p - 1} \right\rbrack},{\xi\lbrack p\rbrack}} \right)} = {\frac{1}{N_{c}}{{BER}\left( {{{H\lbrack p\rbrack},{{output}\left( {{\xi\left\lbrack {p - 1} \right\rbrack},j} \right)}}} \right)}}$where BER(*,*) is computed from:${{BER}\lbrack p\rbrack} = {\frac{1}{N_{s}}{\sum\limits_{k = 1}^{N_{s}}{\phi\left( {\gamma_{k}^{mmse}\lbrack p\rbrack} \right)}}}$as described above for finite state vector quantification (FSVQ).Output(*) is defined as the associated codeword for the j-th outgoingbranch of ξ[p−1], that ends at ξ[p]. At step 108, the optimal codewordis determined from the codebook

represented by B₁ bits and the state ξ[0] of the p=0 subcarrier basedon:${T^{opt}\lbrack p\rbrack} = {\arg\quad{\max\limits_{{T{\lbrack p\rbrack}} \in \quad\mathcal{T}}{{{BER}\lbrack p\rbrack}.}}}$where T^(opt)[p] is the precoded matrix which minimizes the bit errorrate BER[p], at the p-th sub-carrier, and:${{BER}\lbrack p\rbrack} = {\frac{1}{N_{s}}{\sum\limits_{k = 1}^{N_{s}}{{\phi\left( {\gamma_{k}^{mmse}\lbrack p\rbrack} \right)}.}}}$both of which equations are calculated from the same equations used inrecursive feedback. At step 110, the value of p is incremented toevaluate the next (p-th) subcarrier. At step 112, one Viterbi step isexecuted to keep the survivor paths from all possible states on the(p−1)-th subcarrier to all possible states on the p-th subcarrier. Atstep 114, if p is not the last subcarrier, then repeat steps 110 and112. If the last subcarrier has been reached, then at step 116, theoptimal path among all survivor paths is determined. The optimal path isthe path along the trellis which minimizes the average bit error rateBER of the system, such that:$\overset{\_}{BER} = {\sum\limits_{p = 0}^{N_{c} - 1}{{Metric}\left( {{\xi\left\lbrack {p - 1} \right\rbrack},{\xi\lbrack p\rbrack}} \right)}}$At step 118, B₁+(N_(C)−1)B₂ bits are fed back to the transmitter andcorrespond to the indices of the optimum codewords selected. At step120, the transmitter specifies the optimal codewords for all subcarriersbased on the feedback bits.

Referring now to FIGS. 8 and 9, performance degradation with respect toreduced feedback is plotted graphically. The plots apply to MIMO-OFDMusing transmit beamforming on each subcarrier with a beamformingcodebook of size 64 so that B₁ is set to 6 so that per-subcarrierfeedback requires N_(C)B₁=384 bits. To generate the plots, N_(t) is setto 4 and N_(r) is set to 1. B₂ is set to one of 1, 2, 3, 4, and 6. FIG.8 depicts the BER performance vs. Average SNR (Signal-to-noise ratio)for a recursive finite state vector quantification (FSVQ) method, andFIG. 9 depicts BER performance for the trellis-based method. As B₂increases in value, the performance for both methods improves quickly.Note that when B₂=3 in FIG. 9, performance degradation is negligible.The FSVQ method depicted in FIG. 8 works best when the feedbackreduction percentage is small.

FIGS. 10, 11 and 12 plot BER versus Average SNR for transmitbeamforming, spatial multiplexing, and precoded, orthogonal, space-timeblock codes (OSTBC) per subcarrier feedback, respectively, comparing thetrellis feedback reduction method to per-subcarrier feedback and theinterpolation methods. In FIGS. 10 and 12, N_(t) is set to 4 and N_(r)is set to 1. In FIG. 11, N_(t) is set to 4 and N_(r) is set to 2, andN_(s) is set to 2. Without feedback reduction, per-subcarrier feedbackrequires 384 bits when B₁=6 and 128 bits when B₁=2. The trellis methodwith B₁=6 and B₂=2 requires 132 bits. For the interpolation method,N_(g)=16 and P=8, resulting in 144 bits of feedback. As SNR increases inFIGS. 10 and 11, the BER curves of the interpolation-based method leveloff, indicating “diversity loss.” Diversity loss leads to severeperformance degradation at high SNR. The trellis-based methodoutperforms the interpolation-based method at high SNR, and outperformsper-subcarrier feedback by a constant amount (about 1.5 dB).

As demonstrated herein, the present invention overcomes thedisadvantages and shortcomings of the prior art by providing a MIMO-OFDMsystem which employs a finite-rate feedback method utilizing vectorquantization compression. The vector quantization compression techniquesare effective in reducing feedback while maintaining performance atleast in part because subchannel responses across OFDM subcarriers arecorrelated. As will be readily apparent to persons skilled in the art,the present invention has wide applicability and provides significantbenefits to wireless communication systems.

It will be understood that the embodiments described herein are merelyexemplary and that a person skilled in the art may make many variationsand modifications without departing from the spirit and scope of theinvention. All such variations and modifications are intended to beincluded within the scope of the invention as defined in the appendedclaims.

1. A method for improving communications in a MIMO-OFDM system,comprising the steps of: receiving a plurality of symbols from aplurality of sub-carriers at a receiver; selecting a plurality ofindices of codewords corresponding to a codebook of pre-coding weightingmatrices for the sub-carriers based on vector quantization compressionof the codewords; and transmitting the selected indices over a wirelesschannel to a transmitter, wherein the transmitter is adapted to adjustfuture transmissions based upon the selected indices to improvecommunications between the transmitter and the receiver.
 2. The methodof claim 1, wherein said step of selecting indices based on vectorquantization compression of the codewords further includes the step ofselecting the indices based on finite state vector quantization feedbackencoding of the codewords.
 3. The method of claim 1, wherein said stepof selecting indices based on vector quantization compression of thecodewords further includes the step of selecting precoding matrices froma codebook sequentially across the plurality of subcarriers, wherein asubcarrier index, p, is treated as a virtual time index ranging from p=0to p=N_(c)−1, where N_(c) corresponds to a predetermined number ofsubcarriers.
 4. The method of claim 3, wherein said step of selectingindices based on vector quantization compression of the codewords attime index p depends upon the value of a precoding matrix chosen at timep−1.
 5. The method of claim 3, wherein said step of selecting precodingmatrices from a codebook sequentially across the plurality ofsubcarriers further includes the step of: (a) constructing a list of apredetermined total of 2^(B) ² states from a predetermined codebook

represented by B₁ bits, wherein each state in the codebook correspondsto a precoding matrix, T_(i).
 6. The method of claim 5, wherein saidstep of selecting precoding matrices from a codebook sequentially acrossthe plurality of subcarriers further includes the step of: (b)determining an optimal codeword from the codebook

and the state for the p=0 subcarrier.
 7. The method of claim 6, whereinan optimal codeword is determined based on finding an optimal precodingmatrix T^(opt)[p=0] where T^(opt)[p=0] minimizes bit error rate, BER[p],at the p=0 subcarrier.
 8. The method of claim 6, wherein said step ofselecting precoding matrices from a codebook sequentially across theplurality of subcarriers further includes the step of: (c) incrementingthe value of p to evaluate the next (p-th) subcarrier.
 9. The method ofclaim 8, wherein said step of selecting precoding matrices from acodebook sequentially across the plurality of subcarriers furtherincludes the step of: (d) finding the optimal codeword on the p-thsubcarrier using a reduced-size codebook

_(i) which is a subset of

of size 2^(B) ² codewords wherein B₂ is less than B₁, said reduced sizecodebook including codewords that are nearest the optimal codeword ofthe p−1-th subcarrier.
 10. The method of claim 9, wherein step (d) isbased on finding the collection of codewords that minimize the chordaldistance from the codeword chosen at the p−1 th subcarrier.
 11. Themethod of claim 9, wherein said step of selecting precoding matricesfrom a codebook sequentially across the plurality of subcarriers furtherincludes the step of: (e) specifying the state on the p-th subcarrier,wherein the state is one chosen from the reduced size codebook whichminimizes bit error rate.
 12. The method of claim 11, wherein said stepof selecting precoding matrices from a codebook sequentially across theplurality of subcarriers further includes the step of: (f) repeatingsteps (c)-(e) if the p-th subcarrier is not the last subcarrier(N_(C)−1).
 13. The method of claim 12, wherein said step of transmittingthe selected indices over a wireless channel to the transmitter furtherincludes the step of: (g) transmitting B₁+(N_(C)−1)B₂ bits correspondingto the selected indices over the wireless channel to the transmitter.14. The method of claim 13, further including the steps of: (h)receiving the B₁+(N_(C)−1)B₂ bits corresponding to the selected indicesat the transmitter; (i) specifying the optimum codewords for allsubcarriers based on the selected indices; (j) weighting a plurality ofsymbols by a precoding matrix T_(i) based on the optimum codewords; and(k) transmitting the weighted plurality of symbols to the receiver overthe wireless channel.
 15. The method of claim 14, wherein step (j) isbased on one of: per subcarrier beamforming, spatial multiplexing, andprecoded orthogonal space-time block codes (STBC) weighting technique.16. The method of claim 1, wherein said step of selecting a plurality ofindices based on vector quantization compression of the codewordsfurther includes the step of selecting a plurality of indices based ontrellis-based feedback encoding.
 17. The method of claim 16, wherein thestep of selecting indices based on trellis-based feedback encodingfurther includes the step of selecting optimal precoding matrices T_(i)along a trellis for all possible paths at the same time for apredetermined number of subcarriers, N_(c), each subcarrier having adesignation p, p=0 to p=N_(c)−1.
 18. The method of claim 16, wherein anoptimal path through the trellis is chosen to minimize the average biterror rate.
 19. The method of claim 18, wherein the optimal path throughthe trellis is found using the Viterbi algorithm.
 20. The method ofclaim 16, wherein said step of selecting a plurality of indices based ontrellis-based feedback encoding further includes the step of: (a)constructing a trellis by specifying a state transition table where eachstate is characterized by a precoding matrix T_(i) chosen from apredetermined codebook

having 2^(B) ¹ states represented by B₁ bits.
 21. The method of claim20, wherein said step of selecting a plurality of indices based ontrellis-based feedback encoding further includes the step of: (b)defining 2^(B) ² neighbor states from a reduced size codebook of 2^(B) ²codewords where and B₂ is less than B₁, said reduced size codebookincluding codewords that based on descending order of chordal distancerelative to T_(i).
 22. The method of claim 21, wherein said step ofselecting a plurality of indices based on trellis-based feedbackencoding further includes the step of: (c) determining a branch metricfrom the previous state to the current state for all subcarriers. 23.The method of claim 22, wherein said step of selecting a plurality ofindices based on trellis-based feedback encoding further includes thestep of: (d) determining an optimal codeword from the codebook

represented by B₁ bits and the state of the p=0 subcarrier.
 24. Themethod of claim 23, wherein the optimal codeword is determined based onfinding an optimal precoding matrix T^(opt)[p=0] where T^(opt)[p=0]minimizes bit error rate, BER[p], at the p=0 subcarrier.
 25. The methodof claim 23, wherein said step of selecting a plurality of indices basedon trellis-based feedback encoding further includes the step of: (e)incrementing the value of p to evaluate the next (p-th) subcarrier. 26.The method of claim 25, wherein said step of selecting a plurality ofindices based on trellis-based feedback encoding further includes thestep of: (f) executing one Viterbi step that only keeps the survivorpaths from all possible states on the (p−1)-th subcarrier to allpossible states on the p-th subcarrier.
 27. The method of claim 26,wherein said step of selecting a plurality of indices based ontrellis-based feedback encoding further includes the step of: (g)repeating steps (e)-(f) if the p-th subcarrier is not the lastsubcarrier (N_(C)−1).
 28. The method of claim 27, wherein said step ofselecting a plurality of indices based on trellis-based feedbackencoding further includes the step of: (h) finding the optimal pathalong the trellis among all the survivor paths, the optimal pathminimizing average bit error rate.
 29. The method of claim 28, whereinsaid step of transmitting the selected indices over a wireless channelto the transmitter further includes the step of: (i) transmittingB₁+(N_(C)−1)B₂ bits corresponding to the selected indices over thewireless channel to the transmitter.
 30. The method of claim 29, furtherincluding the steps of: (j) receiving the B₁+(N_(C)−1)B₂ bitscorresponding to the selected indices at the transmitter; (k) specifyingthe optimal codewords for all subcarriers based on the selected indices;(l) weighting a plurality of symbols by a precoding matrix T_(i) basedon the optimal codewords; and (m) transmitting the weighted plurality ofsymbols to the receiver over the wireless channel.
 31. The method ofclaim 30, wherein step (l) is based on one of per subcarrierbeamforming, spatial multiplexing, and precoded orthogonal space-timeblock codes (STBC) weighting technique.
 32. A receiver for receiving aplurality of symbols in a MIMO-OFDM system, comprising: a plurality ofantennas for receiving a plurality of symbols from a plurality ofsub-carriers; a MIMO-OFDM de-modulator in signal communication with saidplurality of antennas; a symbol detector in signal communication withsaid MIMO-OFDM de-modulator; a channel estimator in signal communicationwith said MIMO-OFDM de-modulator; a feedback generator in signalcommunication with said channel estimator, said feedback generatorincluding means for selecting a plurality of indices of codewordscorresponding to a codebook of pre-coding weighting matrices for thesub-carriers based on vector quantization compression of the codewords;and means for transmitting the selected indices over a wireless channelto a transmitter.
 33. The receiver of claim 32, wherein said means forselecting a plurality of indices based on vector quantizationcompression of the codewords further includes means for selecting theindices based on finite state vector quantization feedback encoding ofthe codewords.
 34. The receiver of claim 33, wherein said means forselecting a plurality of indices based on vector quantizationcompression of the codewords further includes means for selectingprecoding matrices from a codebook sequentially across the plurality ofsubcarriers.
 35. The receiver of claim 34, wherein said means forselecting a plurality of indices based on a vector quantizationcompression further includes means for selecting a plurality of indicesbased on trellis-based feedback encoding.
 36. The receiver of claim 35,wherein said means for selecting a plurality of indices based ontrellis-based feedback encoding further includes means for selectingoptimal precoding matrices along a trellis for all possible paths at thesame time for a predetermined number of subcarriers.
 37. The receiver ofclaim 36, wherein an optimal path through the trellis is chosen tominimize the average bit error rate.
 38. The receiver of claim 37,wherein the optimal path through the trellis is found using the Viterbialgorithm.